## Introduction

Many a times, while solving Difficult TPA questions, you reach a point where you feel that you do not sufficient information to carry forward or you feel that solving the question will take too long. Whenever you feel this way, always know that there is an inference waiting to be drawn.

Let’s see some examples of inferences that we can draw using Number Properties.

## TPA_032

### Question

This is an official question (not from any of the mocks). Try your hand at it!

### Inference

Now let’s observe how we solved it real quick by drawing inference:

Here is the output of our translation of the dataset:

- Earlier = Sales: Tech = 60:100 = 3:5
- Now = Sales is 12 and Tech is 15

And here is the crucial inference that needs to be drawn at this point of the solution:

- # sales earlier = a multiple of 3 less than 12

And

- # tech earlier = a multiple of 5 less than 15

And from this point onwards, we can work out the possible combinations from the choices to arrive at the final answer.

You may watch the solution from 4:30 mark.

### Summary

We infered that the quantities were multiples of specific numbers using the information about the ratio information given in the dataset. And this paved way further into the solution. This is one of the examples of Difficult TPA Questions. Now, let’s move further.

## TPA_056

### Question

### Inference

Upon simplifying, we arrive at this equation:

To solve this question, we use number properties, specifically related to product of two numbers. Observe how we have take common and arrive at product of an even and an odd number.

And then by comparing the two sides, we infer as shown and find the value of M as 3.

And likewise, we infer and arrive at the value of N as 6.

You may watch the detailed video solution here at 7:36 mark or to watch an alternate solution using the choices, watch from 5:30 mark.

### Summary

In this question, with a single equation with two variables, we converted expression of the sum into product of two numbers and we drew inferences from that product and arrived at the exact values of the two variables. This is another example of Difficult TPA Questions. Let’s see the final one.

## TPA_069

### Question

### Inference

Per the scenario explained in this dataset, processes are repeating at regular intervals based on frequencies specific to each process as shown below:

Whenever you see such repetition, it should automatically infer that LCM is the concept at play. So, to solve this question, this is the crucial inference that you needed to draw. Watch this video at 7:24 to see why this is the case and get a complete clarity on this inference.

### Summary

In this question, once we understood the scenario about repetition of tasks at fixed frequencies, we could infer that this question will require concept of LCM. Hence, we three strategues to solve Difficult TPA Questions.

## Takeaway

While solving TPA questions, when you feel that you are at a point where you cannot go any further, trust your knowledge and identify scope of drawing such inferences. Think of these inferences as the bridge between the point where you get stuck to the next step in your solution.

Inferences = Bridges!

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